NP prob: if n and t are positive integers, what is the great

[sm]

Hello again,

Q: if n and t are positive integers, what is the greatest prime factor of the product nt?
Statement (1): The gratest comon factor of n and t is 5
(2): The least common multiple of n and t is 105.


thanks a bunch!

[Guest]

B is the correct answer but don't know how to attack this prob! thanks

[RonPurewal]

when you take the product of two numbers, all you're doing, in terms of primes, is throwing all the prime factors of both numbers together into one big pool.
therefore, the original question - 'what's the greatest prime factor of the product?' - can be rephrased as,
what's the greatest prime that's a factor of either t or n?[/i]

(1)
because the gcf only tells us which primes are in BOTH t and n. there could be great big fat primes that are factors of only one of them, and they wouldn't show up in the gcf.
insufficient.

(2)
the lcm of two numbers contains EVERY prime that appears in either one of the two numbers (because it's a multiple of both numbers). therefore, whatever is the largest prime factor of the lcm is also the largest prime that goes evenly into either t or n.
sufficient.

--

if you don't realize why the relationships between lcm/gcf and primes, stated above, are what they are, you can just try a few cases and watch the results for yourself. for instance, consider the two numbers 30 (= 2 x 3 x 5) and 70 (= 2 x 5 x 7).
the gcf of these 2 numbers is 10 (= 2 x 5), which doesn't show anything about the presence of the prime factor 7 in one of the numbers.
the lcm of these 2 numbers is 210 (= 2 x 3 x 5 x 7), which contains all of the primes found in either number.